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Questions tagged [parabolic-pde]

A partial differential equation that describes diffusion phenomena.

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Confusion between standard finite element and mass-lumping finite element methods

Consider the following equation, subject to homogeneous Neumann boundary condition. $$ u_t = \Delta u + f(u). $$ The weak formulation is as follows: $$ (u_t,w) = (\Delta u, w) + (f(u),w), \quad \...
Owen Jun's user avatar
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Form of nonlinear diffusion equation

Consider the following nonlinear diffusion problem, $$ \frac{\partial u}{\partial t} = x^{-2}\frac{\partial}{\partial x}\left(x^2 u^4 \frac{\partial u}{\partial x}\right), \quad 0 < x < 1 $$ We ...
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Numerical scheme for the heat equation on the icosahedral hexagonal grid

I have a predefined grid(like this) that is spawned from a regular icosahedradron. It consists of many hexagons and 12 pentagons (corresponding to icosahedradron vertexes). I can tweak the granularity ...
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Convergent Finite Difference Scheme for Parabolic Equation

Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$ where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$. In Strikwerda's book, the ADI scehme \begin{align*} \left( 1 - \frac{...
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Unusual boundary conditions on Matlab

I'm trying to solve the following PDE by Matlab, $$ u_t-\Delta u = 0, \quad \text{in}\quad \Omega\times (0,T) \tag{1} $$ $$ u_t-\Delta_\Gamma u + \partial_\nu u=0,\quad \text{on}\quad \Gamma\times(0,...
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How one could choose the value of viscous coefficient for obtaining stable solution of Burgers' equation?

Burgers' equation is a fundamental PDE used in various fields such as number theory, gas dynamics, heat conduction, elasticity, etc. It is crucial especially for developing numerical models for ...
Loading...'s user avatar
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Modeling First Order Parabolic PDE (Battery Storage Model)

I'm trying to solve the following first order parabolic partial differential equation, \begin{equation*} X \frac{\partial V}{ \partial Q} = -\frac{1}{2} \sigma^2 \frac{\partial^2 V}{\partial X^2} + ...
meraxes's user avatar
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Solving a system of 4 coupled PDEs representing variable diffusivity

I have four partial differential equations representing mass conservation of two compressible fluid phases (marked by subscripts $p1$ and $p2$) in two different continuum media (marked by subscripts $...
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Solvers for stiff initial value ODEs with sparse Jacobian

What ODE solvers are optimized for solving stiff systems with sparse Jacobian? Such systems appear, for instance, when a parabolic PDE is discretized in space using typical finite difference or finite ...
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Effect of reducing flux consistency order at boundary on convergence order

Consider the 1D nonstationary convection-diffusion PDE $$ \begin{alignat}{2} \partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\ f(t) &= \left.\left( a ...
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Oscillations when solving parabolic heat equation with FTCS

I'm wondering if someone could help me out, or point me in a direction of how I can understand the following oscillations that occur when I solve the Porous Medium Equation $$u_t = u_{xx}^{m+1}$$ ...
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Discretizing a parabolic PDE with finite volume method

I want to discretize the following parabolic PDE: $$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\ x\in\Omega \subset \mathbb{R}^2\\ \partial_n u = 0\\ u(t,0) = u_0(x)\ge 0, \alpha(x)>0$$ Given ...
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Accuracy of finite difference method for heat equation on a disk

To study an approximation for the heat equation $$\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2}=f(r,\theta)$$ on the ...
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Growing error from a smooth initial condition for Fisher KPP equation

I'm studying the Fisker-KPP equation on the line (and in $]0, 100[$ numerically): $$ \partial_t u = \Delta_{xx} u + u(1-u) $$ I notice a behavior I don't understand with a smooth initial condition $...
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Implementing initial conditions into the solution domain of a 1-D advection-diffusion equation

I have the following PDE. $\frac{\partial c}{\partial t} + v\frac{\partial c}{\partial x} - D\frac{\partial^2 c}{\partial x^2} = 0 \\$. I have discretized it such that i now have $\frac{dC}{dt} = ...
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explicit scheme stability restriction

Looking at the plain heat equation $u_t=u_{xx}$ the explicit scheme for it would look like the following iteration: $$u_{m,n+1}=\rho u_{m-1,n}+(1-2\rho)u_{m,n}+\rho u_{m+1,n}$$ I noticed this equation ...
Kamil's user avatar
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Stability of two PDEs

I have the following two PDEs which I want to check for stability: $$u_t= u_{xx} , \ u(x,0)=1 , \ u_x(1,t)=-hu^4(1,t) , \ u_x(0,t) = 0 $$ $$u_t = u_{xx}-\sin(x+t)+\cos(x+t) , \ u(x,0)=\cos(x) ,\ u_x(...
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Boundary Conditions for the given PDE

I'm working on the Black-Scholes equation, but I'm pretty new to financial modeling. Right now, I am trying to understand the Black-Scholes PDE. I understand that the Black-Scholes equation is given ...
meraxes's user avatar
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Finite Difference in Polar Co-ordinates

Background Please note that I am duplicating the question on scicomp. I have already asked this in math. I am trying to come up with a scheme in Polar Co-ordinates for the following PDE: PDE I am ...
mod0's user avatar
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How to solve the heat equation using the spectral method (Chebyshev's differentiation matrix), with constant flux boundary condition on both sides?

I am trying to solve a 1d heat equation with a constant flux boundary condition on the right-hand side and a zero flux boundary condition on the left-hand side. I've gained a lot of insight from ...
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The physical meaning of conservative mass in diffusion equation

I am working on 1-D mass transient diffusion in a radial domain (spherical object) using finite volume method. My equation reads $$\frac{\partial C}{\partial t} = D\left[\frac{\partial^2 C}{\partial r^...
Matt's user avatar
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fixed point iteration on DD method

I have to solve the the problem $u_t+\Delta^2u=f(u)$, where $f(u)$ is non-linear, using domain-decomposition method. My approach is first using fixed point iteration on mixed form i.e to say $u^{k+1}...
420's user avatar
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Solving Parabolic PDE using Matlab

I have the following pde (Burger's equation) for $\epsilon>0: u_t+u.u_x=\epsilon.u_{xx}$ and $x\in \mathbb{R},t>0$ and the initial condition: $u(x,0)=\phi(x)=1_{(-\infty,0)}(x)$. I want ...
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Analytic vs discrete understanding of PDE

The PDE I am working with: $$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\ \partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\ \beta(x)>0$$ Integrate the PDE: $$\int_\Omega \partial_t u=\...
sequence's user avatar
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